Alternating-direction Implicit Finite-difference Time-domain Method Research Articles

Kinetic simulations of low-pressure inductively coupled plasma: an implicit electromagnetic PIC/MCC model with the ADI-FDTD method

Low-pressure inductively coupled plasma (ICP) is promising for space electric propulsion. For the first time, an implicit electromagnetic particle-in-cell Monte Carlo collision model based on the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is developed to investigate low-pressure xenon plasma characteristics of a miniature ICP source. The induced simulated electric field is well consistent with that calculated by the finite element method, indicating that this method can provide an accurate estimation of the electromagnetic field. The simulation time step used in the ADI-FDTD method is no longer restricted by the Courant–Friedrichs–Lewy constraints. Compared with the FDTD method, the ADI-FDTD method increases the size of the time step and significantly improves computational efficiency. The method is validated by comparing the simulated and measured electron density and plasma potential profile and reasonable agreement is reached. Therefore, the model is used to investigate the temporal and spatial distribution of plasma properties and the influence of the current amplitude of radio frequency (RF) coil, applied frequency of RF coil and neutral gas pressure on the plasma dynamics in the ionization chamber of a miniature gridded RF ion thruster. To explain the influence of the operating parameters, a concept called ‘the energy relaxation characteristics of electrons in response to the change of electric field’ is proposed and verified. The simulations also find that the oscillation frequency of plasma properties is twice the applied frequency of RF coil. The oscillation characteristics reveal the dynamic energy balance in the ICP. The experiment on the gridded RF ion thruster BHRIT-4 confirms the oscillation by measuring the plasma sheath potential.

Unified Formulation of Leapfrog Complying Divergence Implicit FDTD Method for Lossy Media

The leapfrog alternating direction implicit finite-difference time-domain (ADI-FDTD) method for lossy media has been found to suffer from the non-physical field leakage problem in the non-penetrable structures, which gives rise to inaccuracies in modelling lossy media during scattering analyses. To address such problem, in this paper, we propose a unified formulation of leapfrog complying-divergence implicit (CDI)-FDTD method for lossy media without field leakage problem. Several different lossy schemes to incorporate the conductivity terms of lossy media, including exponential time differencing, averaging and forward-forward schemes are integrated into a unified formulation. The proposed leapfrog CDI-FDTD method not only resolves the problem of field leakage but also has complying divergence. Moreover, since the implicit update procedures in the proposed formulation are in the fundamental form with matrix-operator-free right-hand sides, the floating-point operations (flops) are fewer than that of the conventional leapfrog ADI-FDTD formulation for lossy media. Numerical results are given to demonstrate the advantages of the proposed method over the leapfrog ADI-FDTD method in accuracy, efficiency, complying divergence and resolving the field leakage problem for lossy media.

From Time-Collocated to Leapfrog Fundamental Schemes for ADI and CDI FDTD Methods

The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. For the ADI FDTD method, the time-collocated fundamental schemes are implemented using implicit E-E and E-H update procedures, which comprise simple and concise right-hand sides (RHS) in their update equations. From the fundamental implicit E-H scheme, the leapfrog ADI FDTD method is formulated in conventional form, whose RHS are simplified into the leapfrog fundamental scheme with reduced operations and improved efficiency. For the CDI FDTD method, the time-collocated fundamental scheme is presented based on locally one-dimensional (LOD) FDTD method with complying divergence. The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for CDI FDTD method. Based on their fundamental forms, further insights are given into the relations of leapfrog fundamental schemes for ADI and CDI FDTD methods. The time-collocated fundamental schemes require considerably fewer operations than all conventional ADI, LOD and leapfrog ADI FDTD methods, while the leapfrog fundamental schemes for ADI and CDI FDTD methods constitute the most efficient implicit FDTD schemes to date.

On the Behavior of the LOD-FDTD Method at Dielectric Interfaces

The electric field computed by the locally 1-D finite-difference time-domain (FDTD) method at dielectric interfaces is investigated. To this end, two comprehensive problems are considered, namely a dielectric step in a rectangular waveguide and a dielectric-loaded metallic cavity. We found that, in both problems, the electric field patterns exhibit an unexpected phase shift at the dielectric interface. By contrast, the alternating-direction implicit FDTD method does not present this unphysical behavior.

Memory-optimized shift operator alternating direction implicit finite difference time domain method for plasma

Through introducing the alternating direction implicit (ADI) technique and the memory-optimized algorithm to the shift operator (SO) finite difference time domain (FDTD) method, the memory-optimized SO-ADI FDTD for nonmagnetized collisional plasma is proposed and the corresponding formulae of the proposed method for programming are deduced. In order to further the computational efficiency, the iteration method rather than Gauss elimination method is employed to solve the equation set in the derivation of the formulae. Complicated transformations and convolutions are avoided in the proposed method compared with the Z transforms (ZT) ADI FDTD method and the piecewise linear JE recursive convolution (PLJERC) ADI FDTD method. The numerical dispersion of the SO-ADI FDTD method with different plasma frequencies and electron collision frequencies is analyzed and the appropriate ratio of grid size to the minimum wavelength is given. The accuracy of the proposed method is validated by the reflection coefficient test on a nonmagnetized collisional plasma sheet. The testing results show that the proposed method is advantageous for improving computational efficiency and saving computer memory. The reflection coefficient of a perfect electric conductor (PEC) sheet covered by multilayer plasma and the RCS of the objects coated by plasma are calculated by the proposed method and the simulation results are analyzed.

Using the ADI-FDTD Method to Simulate Graphene-Based FSS at Terahertz Frequency

The alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is modified to simulate a graphene-based frequency selective surface. By using the auxiliary differential equation and Pade fitting method, both the interband and intraband conductivity of the graphene are incorporated into the ADI-FDTD method. The ADI-FDTD method has excellent computational accuracy and its computational efficiency is considerably improved from that of the conventional FDTD method. By using the proposed ADI-FDTD method, a graphene-based frequency selective surface is simulated and analyzed at terahertz frequency. The numerical results show that the graphene can achieve a tunable frequency selective surface through controlling its chemical potential, and the interband conductivity of the graphene has important effect on the performance of the frequency selective surface.

Improved Hybrid Leapfrog ADI-FDTD Method for Simulating Near-Field Coupling Effects Among Multiple Thin Wire Monopole Antennas on a Complex Platform

A hybrid one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method integrated with a thin wire algorithm is developed for simulating the near-field mutual coupling effects among multiple wire antennas on an electrically large warship platform, where induced surface current and electric field distributions are also captured. Its stability analysis, based on von Neumann method, is performed in a semi-analytical way and verified numerically. In order to demonstrate its accuracy and efficiency, the near-field mutual coupling effect between two thin wire antennas is at first investigated. Then, it is applied to accurately characterize complex electromagnetic environment effects on a warship with multiple thin wire antennas, and the results are compared with those obtained by the other commercial simulators.

Modeling and analysis of lightning effects in OPGW cables using the ADI-FDTD method

Current density distribution in optical ground wire cables hit by lightning strokes is numerically calculated and studied in time domain. For this analysis, two models for current source were considered, since atmospheric discharge has two major components which usually damage more severely the cable: the long and short duration current components. The analysis is performed by employing the Alternating Direction Implicit Finite-Difference Time-Domain (ADI-FDTD) method, which the presented formulation was especially developed by the authors to analyze this phenomenon. The convolutional perfectly matched layer (CPML) technique adapted for the ADI method, called ADI-CPML, is also developed in this work for truncating the ADI-FDTD computational domain.

An Efficient Domain Decomposition Parallel Scheme for Leapfrog ADI-FDTD Method

A flexible and universal domain decomposition parallel scheme is proposed for the unconditionally stable finite-difference time-domain (FDTD) method. The leapfrog alternating direction implicit FDTD (ADI-FDTD) method is employed to eliminate the restriction of the Courant–Friedrichs–Lewy stability condition. The proposed domain decomposition parallel implementation of the leapfrog ADI-FDTD method is more flexible with process allocation and requires fewer data communications. A buffer region is introduced to decouple the interactions between neighboring subdomains at each time step. Electromagnetic simulations are presented to demonstrate the applicability, accuracy, and efficiency of the proposed method.

Incorporating interband conductivity into ADI-FDTD method to simulate graphene

The interband conductivity of graphene approximated by using Pade fit method is incorporated into the alternating-direction implicit finite difference time domain (ADI-FDTD) method to simulate graphene. The accuracy of the Pade fit method is verified. The ADI-FDTD method is unconditionally stable, so its computational efficiency is greatly larger than that of FDTD method, which is validated by numerical example. Besides, it shows that the interband conductivity of graphene has important effect on numerical result at high frequency range, especially when chemical potential is small.

Microwave tomography for early breast cancer detection based on the alternating direction implicit finite-difference time-domain method

In microwave tomography (MWT), electric-parameter distributions of the breast can be reconstructed to detect the early-breast-cancer, which has a specific physical explanation and medical diagnostic value. In time-domain, the finite-difference time-domain (FDTD) method is usually applied to these problems. However, due to the constraint of Courant-Friedrich-Levy (CFL) stability condition, the time step should be small enough to well match the small fine cells, which begets an increasing computational cost, such as the central processing unit (CPU) time. For real-time clinical, it is very important and essential to look for efficient methods to improve the computational efficiency. The alternating-direction implicit finite-difference time-domain (ADI-FDTD) method, on the other hand, provides a larger time step than that the CFL stability condition limitation could set. In order to shorten the time of imaging and improve the detection efficiency, the ADI-FDTD method is first used for the early-breast-cancer detection in this paper. MWT for breast cancer detection requires solving nonlinear inverse scattering problems. Most nonlinear inversion algorithms require solving a number of forward scattering problems followed by an optimization procedure. Therefore, we turn the inverse scattering problem into an optimization question according to the least squares criterion. The optimization procedure aims at minimizing the error between measured scattered fields and estimated scattered fields by the forward solver. Nonlinear reconstruction algorithm is used to solve an update for the scattering object properties used in our breast model. This iteration process is repeated until the convergence between the measured and estimated data is obtained. The specific process of the iteration method is divided into two steps: the forward step, which is to solve a forward problem for a scattering object with estimated electrical properties, and the backward step, which is to solve adjoint fields by introducing the Lagrange multiplier penalty function. Both the forward and backward calculations are conducted by using the ADI-FDTD method. The algorithm is evaluated for a two-dimensional (2D) semicircle breast model with tumors. We compare the imaging results obtained by the ADI-FDTD method for various time steps with the results obtained by the conventional FDTD method and the real distribution. The results agree well, the simulation results prove that the imaging time by using this ADI-FDTD method can be reduced to 23% that by the conventional FDTD method. In addition, the simulation results suggest that the ADI-FDTD method can be more efficient if higher resolution is required, thus further enhancing the clinical applicability of MWT.

Scattered Field Leapfrog ADI–FDTD Method for Drude Dispersive Media

This letter presents a new unconditionally stable scattered field (SF) finite-difference time-domain (FDTD) scheme for Drude dispersive medium based on leapfrog alternating direction implicit (ADI) technique. The proposed scheme introduces polarization current density and analytical incident field terms in the leapfrog scheme without adding extra computational complexities except causing minor changes in the coefficients for the central spatial locations. Unconditional stability and accuracy of the scheme are validated by numerical tests. When the Courant number is set to three or a higher value, the scheme can perform faster than the explicit FDTD scheme.

An Effective CFS-PML Implementation for the Cylindrical ADI-FDTD Method

A perfectly matched layer (PML) with complex-frequency-shifted (CFS) constitutive parameters is introduced to the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method in cylindrical grids. The new CFS-PML is based on complex stretching methodology and implemented using bilinear transform. Numerical results show the effectiveness of the proposed implementation.

Broadband modeling surface plasmon polaritons in optically pumped and curved graphene structures with an improved leapfrog ADI-FDTD method

An improved one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is proposed to study surface plasmon polaritons (SPPs) in optically pumped and electrostatic applied curved graphene structures, with their intraband and interband surface conductivities modeled with the vector fitting technique. Such a method can quickly capture the broadband characteristics of SPPs propagating along monolayer, bilayer and trilayer graphene sheets of arbitrary shapes. Numerical stability and accuracy are validated through characterizing field distributions of the SPPs supported in the 180°-curved and spiral graphene sheet waveguides and higher computational efficiency are achieved in comparison with the conventional FDTD method.

Efficient unconditionally stable one‐step leapfrog ADI‐FDTD method with low numerical dispersion

An efficient unconditionally stable one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method based on the controlling parameters is presented. First, three controlling parameters are introduced to the matrices of the Maxwell's equations to decrease the numerical dispersion error, and then the formulation of an efficient one-step leapfrog ADI-FDTD method is derived. Second, the analysis shows that the proposed method is unconditionally stable. Moreover, the numerical dispersion relation of the proposed method is derived analytically. Third, the process of determination of the controlling parameters is shown. Furthermore, the effects of the propagation angle, mesh size, time step and frequency on the dispersion characteristics of the proposed method are also investigated. The result shows that the normalised numerical phase velocity error (NNPVE) and maximum NNPVE of the proposed method are decreased significantly. Finally, two numerical examples are simulated to demonstrate the accuracy and efficiency of the proposed method.

One-Step Leapfrog ADI-FDTD Method in 3-D Cylindrical Grids With a CPML Implementation

A three-dimensional (3-D) unconditionally stable one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method in the cylindrical coordinate system is presented. It is more computationally efficient while preserving the properties of the two-step scheme. By reusing the auxiliary variable, it also uses less memory than the two-step scheme. In contrast with the one-step leapfrog ADI-FDTD method in the Cartesian coordinate system, some implicit equations of the one-step leapfrog ADI-FDTD method in the cylindrical coordinate system are not tridiagonal equations. The Sherman Morrison formula is used to solve them efficiently. To solve open region problems, convolutional perfectly matched layer (CPML) for the one-step leapfrog scheme is proposed. Numerical results are presented to validate them.

High-Order Unconditionally Stable Two-Step Leapfrog ADI-FDTD Methods and Numerical Analysis

High-order unconditionally stable two-step leapfrog alternating direction implicit-finite-difference time-domain (ADI-FDTD) methods in three-dimensional (3-D) domains are presented. Based on the exponential evolution operator (EEO), the Maxwell's equations in a matrix form can be split into four subprocedures first, and then two subprocedures are generated by using the leapfrog scheme. Subsequently, the proposed high-order methods are theoretically proven for unconditional stability, and the numerical dispersion is derived analytically. There are several factors to effect the dispersion: for example, the order of schemes, propagation angle, time step, and mesh size. These four factors are illustrated comprehensively in this paper. Specifically, the normalized numerical phase velocity error (NNPVE) of the proposed schemes is lower than that of the one-step leapfrog ADI-FDTD method. Finally, numerical experiments are presented to demonstrate the validity of the proposed methods. It was found that the proposed second-order method has the same level of relative error as that of the four-step ADI-FDTD method, but it has a higher computational efficiency.

An Error-Reduced ADI-FDTD Algorithm in Debye Media

The alternating direction implicit finite-difference time-domain (ADI-FDTD) method is an unconditionally stable numerical scheme, being proposed to remove stability limitations in conventional FDTD methods. Though the computation efficiency has been improved by ADI-FDTD, significant errors have been observed at large time steps. By compensating truncation errors, a low error ADI-FDTD method in Debye media is proposed based the ER(error reduced)-ADI-FDTD, complete three dimensional equations are derived. Simulation results are anlalyzed and compared with existing methods.

One-step leapfrog ADI-FDTD method for simulating electromagnetic wave propagation in general dispersive media

The one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is reformulated for simulating general electrically dispersive media. It models material dispersive properties with equivalent polarization currents. These currents are then solved with the auxiliary differential equation (ADE) and then incorporated into the one-step leapfrog ADI-FDTD method. The final equations are presented in the form similar to that of the conventional FDTD method but with second-order perturbation. The adapted method is then applied to characterize (a) electromagnetic wave propagation in a rectangular waveguide loaded with a magnetized plasma slab, (b) transmission coefficient of a plane wave normally incident on a monolayer graphene sheet biased by a magnetostatic field, and (c) surface plasmon polaritons (SPPs) propagation along a monolayer graphene sheet biased by an electrostatic field. The numerical results verify the stability, accuracy and computational efficiency of the proposed one-step leapfrog ADI-FDTD algorithm in comparison with analytical results and the results obtained with the other methods.

Analysis of the Divergence Properties for the Three-Dimensional Leapfrog ADI-FDTD Method

Analysis of the divergence properties for the three-dimensional (3-D) leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is presented in this paper. The analysis of divergence properties is based on the Fourier domain plane wave method for dispersion analysis but extended with additional procedure to determine the homogeneous solutions. The singular value decomposition (SVD) is used to obtain the solutions of the homogeneous system in a reliable manner. There are two linearly independent null-modes in the homogeneous system corresponding to the repeated zero singular values. To provide more physical insights, proper linear combinations of the two null-modes are determined as TE and TM modes with respect to the direction of discrete numerical space wavevector. The divergence of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> fields is analyzed by examining the TM mode, while the divergence of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> fields is analyzed by examining the TE mode. Numerical results for the divergence properties of the leapfrog ADI-FDTD method are presented and compared with the conventional ADI-FDTD method, along with numerical experiments to illustrate the existence of divergent waves. The leapfrog ADI-FDTD method is not divergence-free in the source-free region, as is the case for the conventional ADI-FDTD method. Furthermore, both methods have different divergence properties in general.